Dragging center point of cube won't drag all edges to corners with it
The problem is that your faces are non-planar (read this) and therefore need to be converted to triangles to be displayed (Blender will do this without telling us).


Showing us the render wires, make the triangulation apparent.

It is possible to trick Blender into triangulating the nonplanar quads in a different order.


Remove the faces in question and recreate them with triangles.

Select the adjacent faces in the correct order and join them with &#9095; AltJ.

Note, that the correct order is dependant on the vertex index which can be shown only with debug options enabled.


But this is not useful. If you want the area to have an edge explicitly define it with the correct topology.





In the reference video, the same behaviour occurs.